I'm south. I have:
♠️ A 10 5 ♥️ A ♦️ A K Q J 8 ♣️ A Q 10 8
East opened the bidding with 3♥️. What should I bid?
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Typically, with any strong hand (something like 17+) you should begin with a takeout double (if it's available — see below). In this case, your hand is even stronger than that, so you should probably cue bid hearts after almost any bid by partner:
Update: TimLymington points out in the comments that you may have agreed that double here is for penalty. I had not considered this, as I play in the U.S., where the overwhelming majority of pairs play doubles as takeout at least through 4D and often higher.
If double would be for penalty, I recommend doubling and continuing as above except at unfavorable vulnerability. Game for your side is almost certain and slam is a possibility, but it seems difficult to explore for slam when X is not takeout. I expect to beat the opponents by 3 tricks (they'll likely get 6 trump tricks and nothing else), so I'm happy defending when at favorable or equal vulnerability, but the 500 we'll get at unfavorable doesn't compensate us for the 600+ we expect from making our own contract.
In that case, I recommend bidding 3NT, which is a small gamble, but requires very little from partner (the club K guarantees the contract, as does the diamond ten with the spade K, for example).
Using the French system, I should avoid a takeout double which would promise four spades.
As said in answers, 3NT is a gamble (may be risky) !
I checked with my partners and it seems that we agree on a cuebid at 4H which, for us, will promise a very strong hand with at least a 4/5 distribution of cards in clubs and diamonds (and for sure, short in hearts and less than four cards in spades).
If the partner is unable to bid 5C or 5D, then 4NT would be his bid and the contract.
My preferred response is known as a "big" (takeout) double. First of all, you can support play in any suit except hearts, so your double asks your partner to bid his best suit. If partner bids one of the minor suits, you can go to five. If partner bids your "worst" suit, spades, you can correct to four diamonds. If he has a "real" hand (that is, he was bidding three spades with more than xxxx in spades and little or nothing outside), you can easily support a rebid of four spades. (Even a fifth spade or a king from partner would be enough, given your strength).
A perfectly good alternate response is to "cue bid" four hearts (the opposing suit) to show your extra strength, and let partner choose the level (game or slam). This improves your chances of getting to a spade game at the cost of the following:
Doubling allows for the possibility is that partner will pass for penalties with something like Kxx or xxxx in hearts and nothing else. That would be fine, also. You and s/he should be able to take 1-2 heart tricks, and 4-5 in side suits for a 3-4 trick set, which would compensate you for your "derailed" game.
Basically, when you are this strong, you should be able to play the hand (with a decent chance of making game) or extract a large penalty from the opponents. You have such heavy, and balanced, raw strength, that you should let your partner choose where to play (or defend) based on his "shape," that is where he can contribute the most, even without a lot of points. A double allows for all of these possibilities.